| L(s) = 1 | + 3.85i·7-s − 4.85·11-s + 5.85i·13-s + 7.85i·17-s − 2·19-s − 1.85i·23-s + 9.70·29-s − 10.7·31-s + 0.854i·37-s + 8.56·41-s + 5.85i·43-s + 6.70i·47-s − 7.85·49-s + 1.85i·53-s − 7.85·59-s + ⋯ |
| L(s) = 1 | + 1.45i·7-s − 1.46·11-s + 1.62i·13-s + 1.90i·17-s − 0.458·19-s − 0.386i·23-s + 1.80·29-s − 1.92·31-s + 0.140i·37-s + 1.33·41-s + 0.892i·43-s + 0.978i·47-s − 1.12·49-s + 0.254i·53-s − 1.02·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9550274108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9550274108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.85iT - 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 - 5.85iT - 13T^{2} \) |
| 17 | \( 1 - 7.85iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 1.85iT - 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 0.854iT - 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 - 5.85iT - 43T^{2} \) |
| 47 | \( 1 - 6.70iT - 47T^{2} \) |
| 53 | \( 1 - 1.85iT - 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.196028929504709339607250115001, −7.75003499078775810635281458269, −6.61863910565719045173378430568, −6.16129045450115854608044596522, −5.53278390589020298892263242393, −4.71283706551914792769701944958, −4.08199105041108643945922695173, −2.95482109991609222672881014180, −2.27996356424180683821930570853, −1.61542009027964851319180446561,
0.26960896922984068679938693841, 0.839215439299761580986296170765, 2.33019525155835964165707543102, 3.05720642065196295874316943227, 3.73574931834477318907851657219, 4.83609047525579764375611006993, 5.13358877322944366909728541985, 5.98234446991888611173182131110, 6.98920803296040358023646068711, 7.58361607070124797508120870795
